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Question:
Kira is using the figure shown to prove the Pythagorean theorem. She starts by writing the
equation because she knows two equal ways to represent the area
of the shaded region. Which best describes the next steps Kira should take to complete her
proof?
Answer:
The correct option is;
d. Simplify both sides of the equation to get a² + 2·a·b + b² - c² = 2·a·b then subtract 2·a·b and add c² to both sides of the equation
Step-by-step explanation:
Here we have
The area of the larger square = (a + b)²
The area of the middle plain shaded square = c²
∴ Area of the shaded region = (a + b)² - c²
The shaded region consists of for right triangles of base, a and height, b therefore, the area of the shaded region is also the area of the four right triangles = 4 × (1/2 × base × height)
= 4×1/2×a×b = 4(1/2·a·b)
Hence area of the shaded region also = 4(1/2·a·b)
Therefore, (a + b)² - c² = 4(1/2·a·b)
Which is a² + 2·a·b + b² - c² = 4 × 1/2 × a·b = 2·a·b
∴ a² + 2·a·b + b² - c² = 2·a·b
Hence a² + 2·a·b + b² - c² + c² = 2·a·b + c² gives
a² + 2·a·b + b² = 2·a·b + c² gives
Also a² + 2·a·b + b² - 2·a·b = 2·a·b + c² - 2·a·b gives
a² + b² = c² which is Pythagoras theorem